I put this circuit together, as an example. It can be expanded to add a 4th player by adding another button/latch. Only a single counter is necessary as the output of the counter is latched into the respective player's output register. The problem where a player does not push his button is addressed by disabling the output of his register, thus none of his lamps will be lit.
This a good example of a job best handled by a microcontroller, or gate array, as you can see the number of parts and connections involved.